Thus, the length of time to reach steady state depends on the properties of the system and also the initial conditions. Solution for External Forcing. If you have looked at the list of solutions to the equations of motion we derived in the preceding section, you will have discovered that they look horrible.
You will find that the system takes longer to reach steady state. The steady state response of a forced, damped, spring mass system is independent of the initial conditions.
The observation that the system always settles to a steady state has two important consequences. Change the type of forcing, and repeat this test. When analyzing forced vibrations, we almost always neglect the transient response of the system, and calculate only the steady state behavior.
The applet simply calculates the solution to the equations of motion using the formulae given in the list of solutions, and plots graphs showing features of the motion.
You should bear in mind, however, that the steady state is only part of the solution, and is only valid if the time is large enough that the transient term can be neglected.
The transient response depends on everything… Now, reduce the damping coefficient and repeat the test. You will see that, after a while, the solution with the new initial conditions is exactly the same as it was before.
This section summarizes all the formulas you will need to solve problems involving forced vibrations. Unless you have a great deal of experience with visualizing equations, it is extremely difficult to work out what the equations are telling A Java applet posted at http: Note that you can control the properties of the spring-mass system in two ways: The applet will open in a new window so you can see it and read the text at the same time.
You can change the initial velocity too, if you wish. If you look at the solutions to the equations of motion we calculated in the preceding sections, you will see that each solution has the form Following standard convention, we will list only the steady state solutions below.of the input acceleration for any SDOF system over a range of natural frequencies.
1. Take input acceleration signal and filter through SDOF transfer function for natural frequency fn 2. Find maximum amplitude response and plot on Shock Response Spectrum 3.
Increment natural frequency fn. The peak velocity amplitude of a vibrating machine is simply the maximum (peak) vibration speed attained by the machine in a given time period, as shown below.
In contrast to the peak velocity amplitude, the root-mean-square velocity amplitude of a vibrating machine tells us the vibration energy in. Observe that is the amplitude of vibration, and look at the preceding section to find out how the amplitude of vibration varies with frequency, the natural frequency of the system, the damping factor, and the amplitude of the forcing.
It is the most common term used in vibration analysis to describe the frequency of a disturbance. Never forget the 1 cycle / second relationship! Traditional vibration analysis quite often expresses frequency in terms of cycle / minute (cpm).
This is because many pieces of process equipment have running speeds related to revolutions / minute (rpm). the maximum displacement in one direction plus the maximum displacement in the other (includes compression and rarefaction) Greater amplitude will show higher ____ and greater displacement when plotting as a _______.
Vibration of Mechanical Systems 11 2 0 22 d mx kx dt 11 2 2 0 22 m x x k x x or mx kx 0 This is the same equation as we got earlier. Rayleigh’s Method It is a modified energy method. It may be noted that in a conservative system potential energy is maximum when kinetic energy is minimum and vice-versa.Download